Question

How do you factor \[6{x^2} - 5x - 25\] ?

Answer 1

Hint: Here in this question they have given the expression and it is an algebraic expression where it contains variables and constant terms. We have to find the factor \[6{x^2} - 5x - 25\] , so divide it by using variables and constants and hence find the solution.
We are going to use splitting the middle term method to factorise the given quadratic polynomial.

Complete step-by-step answer:
The equation \[6{x^2} - 5x - 25\] is a quadratic equation and we get two factors on factorization. This equation is an algebraic equation which contains both constant term and variable term. To find the factor for the equation \[6{x^2} - 5x - 25\] we have a rule and it is a sum-product rule. To find the factors let we equate it to zero \[6{x^2} - 5x - 25 = 0\]
{Generally, the sum-product rule of the equation is
Consider the equation in general \[a{x^2} + bx + c = 0\] where the product of a and c is written as sum of b. The numbers should satisfy the b by applying the sum rule. Here the sign convention plays an important role.}
Now consider the equation \[6{x^2} - 5x - 25 = 0\] , here a= 6 b=-5 and c=-25
The product of ac is -150
The factors of -150 is 10 and -15 and the factors of -150 is -10 and 15
Now we will check which factors on the sum rule will satisfy the b
If we apply the sum rule to 10 and -15 the answer is -5
If we apply the sum rule to -10 and 15 the answer is 5
So we will consider 10 and -15 as factors of -150
Therefore the equation can be written as
 \[6{x^2} - 5x - 25 = 0\]
 \[ \Rightarrow 6{x^2} + 10x - 15x - 25 = 0\]
Rearrange the terms in the equation
 \[ \Rightarrow 6{x^2} + 10x - 15x - 25 = 0\]
Let we take 2x as common in \[6{x^2} + 10x\] and -5 as common in \[ - 15x - 25\] and the equation is written as
 \[ \Rightarrow 2x(3x + 5) - 5(3x + 5) = 0\]
Take (3x+5) as common in the above equation we have
 \[(3x + 5)(2x - 5) = 0\]
Therefore the factors of \[6{x^2} - 5x - 25 = 0\] is \[(3x + 5)(2x - 5) = 0\]
So, the correct answer is “ \[(3x + 5)(2x - 5) = 0\] ”.

Note: The sum product rule is used to find the factors and will obtain the solution for the question. In general the sum product is given as the equation in general \[a{x^2} + bx + c = 0\] where the product of a and c is written as sum of b. The numbers should satisfy the b by applying the sum rule. Here the sign convention plays an important role. The sign conventions should be known to satisfy the equation.